Measurements of long-lived isotopes in meteoritic data indicate the flux of Galactic cosmic rays (GCRs) has been constant for several Myr, but these measurements have relative systematic uncertainties that may exceed 30%. By using radiocarbon monoxide ($^{14}$CO) from ice cores in interior East Antarctica extracted from carbon monoxide trapped in the Antarctic ice sheet at Dome C, we can reconstruct changes in the GCR flux over a shorter timescale ($\sim$7 kyr) but with much higher precision than previously achieved. This approach requires accurate characterization of the muon fluxes and muogenic $^{14}$CO production rates in ice. We present a model of the relationship between cosmic rays at the top of the atmosphere and $^{14}$CO formed in Antarctic ice using atmospheric and in-ice particle cascades, measurements of muon interaction cross-sections, and ice flow models to track the formation and accumulation of $^{14}$C. We describe applications of the model to anticipated $^{14}$CO measurements from Dome C, Antarctica and forecast constraints on the recent history of the GCR flux.